At Just the Right Time, God Sent His Son
Galatians 4:1-5

Even though the child will someday inherit and control all the riches of the family, before he becomes a legal adult he is treated no differently than a slave (Gal 4:1-2)

Child (nepios) = one without understanding, an infant or young child.
This verse acknowledges the fact that this child is really the heir to the inheritance. However, it describes an important limitation in the timing for actually taking control of this inheritance.
Although the child is the owner (kurios = "lord") of everything, he does not yet have actual control over anything. In this respect he is no different than one of the slaves in the household. He, too, is under the supervision of guardians and managers that are appointed by the father (who is the present "lord" of the inheritance).
From the perspective of possessing the inheritance, the heir is strictly controlled (like a slave) UNTIL a specific future time.

Now Paul applies the illustration to us: just as in the real-life analogy of the Roman child and heir, God determined that all of mankind would be held in bondage until a specific time which He had set in His eternal plan.
A general set of worldly principles served as our "guardian." The elemental things of the world is a phrase that broadens the definition of "guardian" to include the Jewish Law as well as Gentile laws, cultural rituals, folk wisdom, common sense, conscience, and any man-made religious system.
All of mankind has been held in bondage to these kinds of things until the appointed time set by God.

But... contrasts the condition described in Gal 4:1-3 with the change that took place when God's appointed time actually arrived.
This verse indicates that there was a specific moment in the history of the world when all of God's preparations for the salvation of mankind came to completion - everything fell into place just as He planned.
1. The time was right religiously - the Jews had completed the Old Testament canon, and they had developed the synagogue system to function as schools, courts, and for worship.
2. The time was right culturally - there was a common language (koine Greek) and a common cultural background across the known world.
3. The time was right politically - Rome had instituted a common system of law that resulted in the Pax Romana (Roman peace), and they established an extensive system of roads and trade routes which made travel relatively easy and safe.
When everything was in place, God sent forth His Son - Jesus Christ the Messiah and Savior of mankind. Jesus was born exactly as predicted by Old Testament prophecy - born of a Jewish virgin and born under the Law.

This verse gives us the ultimate purpose for the birth of Jesus in Bethlehem.
Redeem (exagorazo) = to buy out of the slave market. Christ came to give Himself as the price of redemption for those He desires to purchase out of the slave market.
Even in the Roman culture of that time, a slave could be redeemed and could become a free person in Roman society. But not only did Christ purchase freedom for his chosen ones, He went far beyond that.
adoption as sons (huiothesias) = literally, placing into sonship - this was the act of giving the status of sonship to someone who was not one's natural child.
Now that we are free and eligible for adoption, God went the whole way and actually adopted us into His family and made us heirs to His entire fortune! This went far beyond mere "freedom" - it was much more than we could ever have asked for or expected.